Roger Bagula <tftn at earthlink.net> wrote in message news:<ckajb5$m4j$1 at smc.vnet.net>...
> If you take the first and last term away from a binomial polynomial and
> set the result equal to zero,
> you get a number of strange roots.
> This method allows you to plot such roots.
> I didn't know it would work when I wrote it up,
> but I plan to use it in the future
> on some other polynomial root structures.
>
> (* root group where x^q+1=(x+1)^q: binomial expansion without x^q and 1*)
> digits=21
> s[q_]=Sum[(q!/((q-k)!*k!))*x^(q-k),{k,1,q-1}]
> ExpandAll[s[2]]
> ExpandAll[s[3]]
> a=Flatten[Table[x/. NSolve[s[n]==0,x],{n,2,digits}]];
> a0=Floor[Abs[a]]
> Dimensions[a][[1]]
continued...
b = Table[{Re[a[[n]]], Im[a[[n]]]}, {n, 1, Dimensions[a][[1]]}];
ChopEnds = ListPlot[b, PlotRange -> {{-1, 2}, {-1, 1}}];
central = ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 2 Pi}];
displ = ParametricPlot[{Cos[t] - 1, Sin[t]}, {t, 0, 2 Pi}];
Show[ChopEnds, central, displ];
Hi Roger,
The roots are neatly herded onto unit circles centered on (0,0) and
(-1,0), (except one point (-0.5, +/-1), as may be expected for complex
roots of z^(1/n),(z+1)^(1/n) somehow with only negative real parts,
|x|<1 .
Regards. Nara